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No, Bear's head is operated by the puppeteer's right hand.
Bear's right arm/hand is stuffed and rigged to move up and down in opposition to his left arm, to make for more natural-looking arm movements when he walks or dances. Note how Bear is left-handed on the show; he seems to do most things with his left hand. This is because the puppeteer's right hand is busy moving the head and mouth.
(In a similar fashion, most Muppets are left-handed, because their puppeteers are right-handed, so the right hand is almost always used to operate the puppet's head.)